The expression on the up is largely used in the UK and means to improve or to move towards a better state. This expression is a shortened versing of 'on the up and up' meaning to become more successful. Something which is on the increase is becoming greater more common or more frequent. When we talk about something being on the increase it refers to something progressively becoming greater in size amount number or value over time. Something which is on the rise can refer to something becoming more numerous or which is experiencing an increase or becoming more successful.

1. With violent crime on the rise, the government is coming under increasing pressure to put more police officers on the street.
2. The water levels in the river are on the rise we are afraid that it might flood.
3. Employment is on the up, it's a great time to be looking for a new job.

1. A new report has shown that stress in the workplace is on the increase due to the demands being put on workers in what has become a highly competitive labour market.
2. Gary's career has been on the up ever since he took that communications course. The skills he learnt there really helped to develop his managerial capabilities.